Kovalchuk is one of more than 20 Russians barred from entering the United States or holding assets there under sanctions imposed over Russia's annexation of Crimea from Ukraine, which Western leaders say is illegal.
Putin said on Friday he would open an account at Bank Rossiya. Kovalchuk, who was also hit by sanctions personally, said the bank had been "flooded with people" who had heard about Putin's plan and wanted to do the same.
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"What happens the day after a bank gets in trouble? Everyone pulls their money out. The morning after (the sanctions were announced), everyone was ready to see depositors take their money out. Journalists were standing by. Nobody came," he said.
"And when in the afternoon it was announced ... that Putin would open an account, the bank was flooded with people. All kinds of people—famous people, ordinary people. They came to open accounts in Bank Rossiya."
He did not produce any figures to back up his suggestion in an interview with Dmitry Kiselyov, a Kremlin-favoured television anchor targeted by similar moves by the European Union.
Described by the U.S. government as "the personal banker for senior Russian officials, Kovalchuk is the largest shareholder in Bank Rossiya, which is based in St Petersburg.
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In a warning to other tycoons, he said "people intuitively understand which side of the barricade a business is on."
"You can have an apartment abroad or a villa on the (French) Riviera. Fine. The question is, where is your home? And one's home is not just money. Where is your family, where do your children go to school, where do they work?" he added.
In what could be a reference to Roman Abramovich, the London-based owner of Chelsea soccer club, he said: "And what sports team do you sponsor? Businesses are different—one might sponsor, say, a serious soccer team in the premier league, another a sandlot (unorganized) team. That's not important —the question is, where is the team—here or outside your country?"
Kovalchuk took a similar stance to others hit by sanctions by wearing them as a badge of honour and shrugging off their impact. Putin has mocked their potential impact but senior officials have said Russia will respond with reciprocal moves.
The latest U.S. sanctions were targeted at Putin's inner circle but the president himself has not been hit by sanctions.
Kovalchuk said Russia must decide how much it is willing to sacrifice for the sake of approval and acceptance by the West, suggesting it may not need the West.
"We need to answer for ourselves: to what degree we are ready to integrate into another structure, and to what degree it suits us," he said.